4.13 D Classes for Transformation Monoids

In addition to the usual components of a D class record, the record representing the D class dClass of s in a transformation monoid can have the following components. They are created by the function SchutzenbergerGroup (see SchutzenbergerGroup) which is called whenever the size, the list of elements of dClass, or a membership test in dClass is asked for.

schutzenbergerGroup:

set to the Schaccent127utzenberger group of dClass as a permutation group on the set of images of SchutzenbergerGroup for Transformation Monoids). H:
set to the H class of dClass.representative.

L:

set to the L class of dClass.representative.

R:

set to the R class of dClass.representative. rCosets:
contains a list of (right) coset representatives of the Schaccent127utzenberger group of dClass in the Schaccent127utzenberger group of the R class dClass.R.

The following functions have a special implementation in terms of these components.

Size( dClass )

returns the size of the D class dClass. This function calls SchutzenbergerGroup and determines the size of dClass in terms of the sizes of the resulting group and the Schaccent127utzenberger groups of the R class dClass.R and the L class dClass.L.

Elements( dClass )

returns the set of elements of the D class dClass. This function calls SchutzenbergerGroup and determines the set of elements of dClass as the union of cosets of the Schaccent127utzenberger group of the L class dClass.L determined through the multipliers dClass.rCosets and dClass.R.rMults.

x in dClass

returns true if x is an element of the D class dClass and false otherwise. This function calls SchutzenbergerGroup and tests whether the quotient of the representative of dClass and a suitable translate of x can be found in one of the cosets of the Schaccent127utzenberger group of the L class dClass.L determined by the list dClass.rCosets.

HClasses( dClass )

returns the list of H classes contained in the D class dClass.

LClasses( dClass )

returns the list of L classes contained in the D class dClass.

RClasses( dClass )

returns the list of R classes contained in the D class dClass.

   
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